Solution
= Solution
Use the separation definition in (i). If $A=\bigcup_i A_i$ had a separation $A=U\cup V$, the intersections with each <connected> $A_i$ show that $A_i$ lies wholly in one side. Choose one nonempty $A_{i_0}$; every other $A_i$ intersects it and must lie on that same side. Thus the other side is empty, a contradiction. If the union is empty it is <connected> by definition. Therefore $\boxed{\bigcup_iA_i\text{ is connected}}$.